Let denote the set of log canonical thresholds of pairs , with a nonsingular variety of dimension , and a nonempty closed subscheme of . Using non-standard methods, we show that every limit of a decreasing sequence in lies in , proving in this setting a conjecture of Kollár. We also show that is closed in ; in particular, every limit of log canonical thresholds on smooth varieties of fixed dimension is a rational number. As a consequence of this property, we see that in order to check Shokurov’s ACC Conjecture for all , it is enough to show that is not a point of accumulation from below of any . In a different direction, we interpret the ACC Conjecture as a semi-continuity property for log canonical thresholds of formal power series.
Dans cet article, nous analysons les ensembles de seuils log canoniques de paires , où est une variété lisse de dimension , et est un sous-schéma fermé non-vide de . En employant des méthodes non-standard, nous montrons que chaque limite d’une suite strictement décroissante de appartient à l’ensemble (ce résultat a été conjecturé par J. Kollár dans ses travaux sur le sujet). Nous montrons également que l’ensemble est fermé dans , et en déduisons que les valeurs adhérentes de l’ensemble des seuils log canoniques des pairs sont rationnelles, si la dimension de est majorée. Une autre conséquence de nos résultats concerne la conjecture ACC de Shokurov pour les . En effet, nous montrons qu’elle est une conséquence de l’énoncé suivant : pour tout , la valeur ne peut pas être obtenue comme limite d’une suite strictement croissante de nombres contenus dans . Dans une autre perspective, nous interprétons la conjecture ACC comme une propriété de semi-continuité de seuils log canoniqes des séries formelles.
Keywords: log canonical threshold, multiplier ideals, ultrafilter, resolution of singularities
Mot clés : seuils log canoniques, idéaux multiples, ultra-filtres, résolution de singularités
@article{ASENS_2009_4_42_3_491_0, author = {de Fernex, Tommaso and Mustaț\u{a}, Mircea}, title = {Limits of log canonical thresholds}, journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure}, pages = {491--515}, publisher = {Soci\'et\'e math\'ematique de France}, volume = {Ser. 4, 42}, number = {3}, year = {2009}, doi = {10.24033/asens.2100}, mrnumber = {2543330}, zbl = {1186.14007}, language = {en}, url = {http://www.numdam.org/articles/10.24033/asens.2100/} }
TY - JOUR AU - de Fernex, Tommaso AU - Mustață, Mircea TI - Limits of log canonical thresholds JO - Annales scientifiques de l'École Normale Supérieure PY - 2009 SP - 491 EP - 515 VL - 42 IS - 3 PB - Société mathématique de France UR - http://www.numdam.org/articles/10.24033/asens.2100/ DO - 10.24033/asens.2100 LA - en ID - ASENS_2009_4_42_3_491_0 ER -
%0 Journal Article %A de Fernex, Tommaso %A Mustață, Mircea %T Limits of log canonical thresholds %J Annales scientifiques de l'École Normale Supérieure %D 2009 %P 491-515 %V 42 %N 3 %I Société mathématique de France %U http://www.numdam.org/articles/10.24033/asens.2100/ %R 10.24033/asens.2100 %G en %F ASENS_2009_4_42_3_491_0
de Fernex, Tommaso; Mustață, Mircea. Limits of log canonical thresholds. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 42 (2009) no. 3, pp. 491-515. doi : 10.24033/asens.2100. http://www.numdam.org/articles/10.24033/asens.2100/
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